relecture abstract et intro et début de la section du l'async

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 8e39b0786c0e08fbada6a6cdc128790199773065..cea0a410d588e278172f8b1c4f6c89eda66fda5a 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -88,11 +88,9 @@ analysis of simulated grid-enabled numerical iterative algorithms}
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
 }
 
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
 }
 

-%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non
-Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email:
-l.zianekhodja@ulg.ac.be
+%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be

 
 

-\begin{abstract}   The behavior of multicore applications is always a challenge
+\begin{abstract}   The behavior of multi-core applications is always a challenge

 to predict, especially with a new architecture for which no experiment has been
 performed. With some applications, it is difficult, if not impossible, to build
 accurate performance models. That is why another solution is to use a simulation
 to predict, especially with a new architecture for which no experiment has been
 performed. With some applications, it is difficult, if not impossible, to build
 accurate performance models. That is why another solution is to use a simulation


@@ -103,7 +101,7 @@ applications.
 
 In this paper, we focus our attention on two parallel iterative algorithms based
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
 
 In this paper, we focus our attention on two parallel iterative algorithms based
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.

-These algorithms  are used to  solve libear  systems. Two different  variantsof
+These algorithms  are used to  solve libear  systems. Two different  variants of

 the Multisplitting are studied: one  using synchronoous  iterations and  another
 one  with asynchronous iterations. For each algorithm we have  tested different
 parameters to see their influence.  We strongly  recommend people  interested
 the Multisplitting are studied: one  using synchronoous  iterations and  another
 one  with asynchronous iterations. For each algorithm we have  tested different
 parameters to see their influence.  We strongly  recommend people  interested


@@ -117,13 +115,12 @@ their  applications  using  a simulation tool before.
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 

-\keywords{Multisplitting algorithms, Synchronous and asynchronous
-iterations, SimGrid, Simulation}
+\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}

 
 \maketitle
 
 
 \maketitle
 

-\section{Introduction}  The use of multi-core architectures for solving large
-scientific problems seems to  become imperative  in  a  lot  of  cases.
+\section{Introduction}  The use of multi-core architectures to solve large
+scientific problems seems to  become imperative  in  many situations.

 Whatever the scale of these architectures (distributed clusters, computational
 grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
 complex parallel applications operating on a large amount of data.
 Whatever the scale of these architectures (distributed clusters, computational
 grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
 complex parallel applications operating on a large amount of data.


@@ -134,8 +131,7 @@ are often very important. So, in this context it is difficult to optimize a
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,

-code debugging, ability to obtain results quickly,~\ldots at the condition that
-the simulation results are in education with the real ones.
+code debugging, ability to obtain results quickly,~\ldots. In counterpart, the simulation results need to be consistent with the real ones.

 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite
 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite


@@ -150,7 +146,7 @@ from its neighbors. We say that the iteration computation follows a synchronous
 scheme. In the asynchronous scheme a task can compute a new iteration without
 having to wait for the data dependencies coming from its neighbors. Both
 communication and computations are asynchronous inducing that there is no more
 scheme. In the asynchronous scheme a task can compute a new iteration without
 having to wait for the data dependencies coming from its neighbors. Both
 communication and computations are asynchronous inducing that there is no more

-idle times, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
+idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}.

 This model presents some advantages and drawbacks that we detail in
 section~\ref{sec:asynchro} but even if the number of iterations required to
 converge is generally  greater  than for the synchronous  case, it appears that
 This model presents some advantages and drawbacks that we detail in
 section~\ref{sec:asynchro} but even if the number of iterations required to
 converge is generally  greater  than for the synchronous  case, it appears that


@@ -158,15 +154,15 @@ the asynchronous  iterative scheme  can significantly  reduce  overall execution
 times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
 for more details).
 
 times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
 for more details).
 

-Nevertheless, in both cases (synchronous or asynchronous) it is very time
-consuming to find optimal configuration and deployment requirements  for a given
-application on a given multi-core architecture. Finding good resource
-allocations policies under varying CPU power, network speeds and  loads is very
-challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
-problematic is even more difficult for the asynchronous scheme  where variations
-of the parameters of the execution platform can lead to very different number of
-iterations required to converge and so to very different execution times. In
-this challenging context we think that the use of a simulation tool can greatly
+Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
+consuming to find optimal configuration  and deployment requirements for a given
+application  on   a  given   multi-core  architecture.  Finding   good  resource
+allocations policies under  varying CPU power, network speeds and  loads is very
+challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
+problematic is  even more difficult  for the  asynchronous scheme where  a small
+parameter variation of the execution platform can lead to very different numbers
+of iterations to reach the converge and so to very different execution times. In
+this challenging context we think that the  use of a simulation tool can greatly

 leverage the possibility of testing various platform scenarios.
 
 The main contribution of this paper is to show that the use of a simulation tool
 leverage the possibility of testing various platform scenarios.
 
 The main contribution of this paper is to show that the use of a simulation tool


@@ -178,7 +174,7 @@ algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
 solver~\cite{saad86} in synchronous mode. The obtained results on different
 simulated multi-core architectures confirm the real results previously obtained
 on non simulated architectures.  We also confirm  the efficiency  of the
 solver~\cite{saad86} in synchronous mode. The obtained results on different
 simulated multi-core architectures confirm the real results previously obtained
 on non simulated architectures.  We also confirm  the efficiency  of the

-asynchronous  multisplitting algorithm  comparing to the synchronous  GMRES. In
+asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. In

 this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
 this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no


@@ -196,6 +192,25 @@ concluding remarks and perspectives.
 \section{The asynchronous iteration model}
 \label{sec:asynchro}
 
 \section{The asynchronous iteration model}
 \label{sec:asynchro}
 

+Asynchronous iterative methods have been  studied for many years theorecally and
+practically. Many methods have been considered and convergence results have been
+proved. These  methods can  be used  to solve in  parallel fixed  point problems
+(i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,
+asynchronous iterations  methods can be used  to solve, for example,  linear and
+non-linear systems of equations or optimization problems, interested readers are
+invited to read~\cite{BT89,bahi07}.
+
+Before  using  an  asynchronous  iterative   method,  the  convergence  must  be
+studied. Otherwise, the  application is not ensure to reach  the convergence. An
+algorithm that supports both the synchronous or the asynchronous iteration model
+requires very few modifications  to be able to be executed  in both variants. In
+practice, only  the communications and  convergence detection are  different. In
+the  synchronous   mode,  the  iterations   are  synchronized  whereas   in  the
+asynchronous  one,  they are  not.   It  should  be  noticed that  non  blocking
+communications can be used in  both modes. Concerning the convergence detection,
+synchronous variants  can use  a global  convergence procedure  which acts  as a
+global synchronization point. In the asynchronous model, the convergence dectection is more tricky as it must not synchronize all the processors. Interested readers can consult~\cite{myBCCV05c,bahi07,ccl09:ij}. 
+

 \section{SimGrid}
  \label{sec:simgrid}
 
 \section{SimGrid}
  \label{sec:simgrid}
 


@@ -290,10 +305,10 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 

-One of our objectives when simulating the application in SIMGRID is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in SIMGRID simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.
+One of our objectives when simulating the application in Simgrid is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in Simgrid simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.

 
 
 
 

-\paragraph{SIMGRID Simulator parameters}
+\paragraph{Simgrid Simulator parameters}

 
 \begin{itemize}
        \item hostfile: Hosts description file.
 
 \begin{itemize}
        \item hostfile: Hosts description file.


@@ -315,7 +330,7 @@ In addition, the following arguments are given to the programs at runtime:
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 

-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determine the computational power (here 19GFlops) of the simulator host machine.
+At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.

 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%


@@ -324,7 +339,7 @@ At last, note that the two solver algorithms have been executed with the Simgrid
 \label{sec:expe}
 
 
 \label{sec:expe}
 
 

-\subsection{Setup study and Methodology}
+\subsection{Study setup and Simulation Methodology}

 
 To conduct our study, we have put in place the following methodology
 which can be reused for any grid-enabled applications.
 
 To conduct our study, we have put in place the following methodology
 which can be reused for any grid-enabled applications.


@@ -335,8 +350,8 @@ have been chosen for the study in this paper. \\
 
 \textbf{Step 2} : Collect the software materials needed for the
 experimentation. In our case, we have two variants algorithms for the
 
 \textbf{Step 2} : Collect the software materials needed for the
 experimentation. In our case, we have two variants algorithms for the

-resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, SIMGRID simulator has been chosen to simulate the behaviors of the
-distributed applications. SIMGRID is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
+resolution of the 3D-Poisson problem: (1) using the classical GMRES (Algo-1); (2) and the multisplitting method (Algo-2). In addition, Simgrid simulator has been chosen to simulate the behaviors of the
+distributed applications. Simgrid is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\

 
 \textbf{Step 3} : Fix the criteria which will be used for the future
 results comparison and analysis. In the scope of this study, we retain
 
 \textbf{Step 3} : Fix the criteria which will be used for the future
 results comparison and analysis. In the scope of this study, we retain


@@ -344,20 +359,19 @@ in one hand the algorithm execution mode (synchronous and asynchronous)
 and in the other hand the execution time and the number of iterations of
 the application before obtaining the convergence. \\
 
 and in the other hand the execution time and the number of iterations of
 the application before obtaining the convergence. \\
 

-\textbf{Step 4 }: Setup up the different grid testbeds environment
+\textbf{Step 4 }: Set up the different grid testbed environments

 which will be simulated in the simulator tool to run the program. The
 following architecture has been configured in Simgrid : 2x16 - that is a
 grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8,
 4x16, 8x8 and 2x50. The network has been designed to operate with a
 which will be simulated in the simulator tool to run the program. The
 following architecture has been configured in Simgrid : 2x16 - that is a
 grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8,
 4x16, 8x8 and 2x50. The network has been designed to operate with a

-bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8E-6
-microseconds (resp. 5E-5) for the intra-clusters links (resp.
+bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8.10$^{-6}$
+microseconds (resp. 5.10$^{-5}$) for the intra-clusters links (resp.

 inter-clusters backbone links). \\
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations in varying the key parameters, especially
 the CPU power capacity, the network parameters and also the size of the
 inter-clusters backbone links). \\
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations in varying the key parameters, especially
 the CPU power capacity, the network parameters and also the size of the

-input matrix. Note that some parameters should be fixed to be invariant to allow the
-comparison like some program input arguments. \\
+input matrix. Note that some parameters like some program input arguments should be fixed to be invariant to allow the comparison. \\

 
 \textbf{Step 6} : Collect and analyze the output results.
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 


@@ -383,7 +397,7 @@ data from a source and the final time the destination have finished to
 receive it. Upon the network characteristics, another impacting factor
 is the application dependent volume of data exchanged between the nodes
 in the cluster and between distant clusters. Large volume of data can be
 receive it. Upon the network characteristics, another impacting factor
 is the application dependent volume of data exchanged between the nodes
 in the cluster and between distant clusters. Large volume of data can be

-transferred in transit between the clusters and nodes during the code
+transferred and transit between the clusters and nodes during the code

 execution.
 
  In a grid environment, it is common to distinguish in one hand, the
 execution.
 
  In a grid environment, it is common to distinguish in one hand, the


@@ -403,7 +417,7 @@ synchronous mode}
 In the scope of this paper, our first objective is to demonstrate the
 Algo-2 (Multisplitting method) shows a better performance in grid
 architecture compared with Algo-1 (Classical GMRES) both running in
 In the scope of this paper, our first objective is to demonstrate the
 Algo-2 (Multisplitting method) shows a better performance in grid
 architecture compared with Algo-1 (Classical GMRES) both running in

-\textbf{\textit{synchronous mode}}. Better algorithm performance
+\textit{synchronous mode}. Better algorithm performance

 should means a less number of iterations output and a less execution time
 before reaching the convergence. For a systematic study, the experiments
 should figure out that, for various grid parameters values, the
 should means a less number of iterations output and a less execution time
 before reaching the convergence. For a systematic study, the experiments
 should figure out that, for various grid parameters values, the


@@ -416,7 +430,7 @@ and our comments.\\
 
 
 \textit{3.a Executing the algorithms on various computational grid
 
 
 \textit{3.a Executing the algorithms on various computational grid

-architecture scaling up the input matrix size}
+architecture and scaling up the input matrix size}

 \\
 
 % environment
 \\
 
 % environment


@@ -437,8 +451,7 @@ Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 

-The results in figure 3 show the non-variation of the number of
-iterations of classical GMRES for a given input matrix size; it is not
+In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration the non-variation of the number of iterations of classical GMRES for a given input matrix size; it is not

 the case for the multisplitting method.
 
 %\begin{wrapfigure}{l}{100mm}
 the case for the multisplitting method.
 
 %\begin{wrapfigure}{l}{100mm}


@@ -450,15 +463,13 @@ the case for the multisplitting method.
 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-Unless the 8x8 cluster, the time
-execution difference between the two algorithms is important when
+The execution time difference between the two algorithms is important when

 comparing between different grid architectures, even with the same number of
 processors (like 2x16 and 4x8 = 32 processors for example). The
 experiment concludes the low sensitivity of the multisplitting method
 comparing between different grid architectures, even with the same number of
 processors (like 2x16 and 4x8 = 32 processors for example). The
 experiment concludes the low sensitivity of the multisplitting method

-(compared with the classical GMRES) when scaling up to higher input
-matrix size.
+(compared with the classical GMRES) when scaling up the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs 40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.

 
 

-\textit{\\3.b Running on various computational grid architecture\\}
+\textit{\\3.b Running on two different speed cluster inter-networks\\}

 
 % environment
 \begin{footnotesize}
 
 % environment
 \begin{footnotesize}


@@ -485,11 +496,11 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on

-a speed inter- cluster network (N1) and a less performant network (N2).
+a speed inter- cluster network (N1) and also on a less performant network (N2).

 Figure 4 shows that end users will gain to reduce the execution time
 for both algorithms in using a grid architecture like 4x16 or 8x8: the
 performance was increased in a factor of 2. The results depict also that
 Figure 4 shows that end users will gain to reduce the execution time
 for both algorithms in using a grid architecture like 4x16 or 8x8: the
 performance was increased in a factor of 2. The results depict also that

-when the network speed drops down, the difference between the execution
+when the network speed drops down (12.5\%), the difference between the execution

 times can reach more than 25\%.
 
 \textit{\\3.c Network latency impacts on performance\\}
 times can reach more than 25\%.
 
 \textit{\\3.c Network latency impacts on performance\\}


@@ -550,11 +561,8 @@ Table 4 : Network bandwidth impact \\
 
 
 
 
 
 

-The results of increasing the network bandwidth depict the improvement
-of the performance by reducing the execution time for both of the two
-algorithms (Figure 6). However, and again in this case, the multisplitting method
-presents a better performance in the considered bandwidth interval with
-a gain of 40\% which is only around 24\% for classical GMRES.
+The results of increasing the network bandwidth show the improvement
+of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.

 
 \textit{\\3.e Input matrix size impacts on performance\\}
 
 
 \textit{\\3.e Input matrix size impacts on performance\\}
 


@@ -599,7 +607,7 @@ same test has been done with the grid 2x16 getting the same conclusion.
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline

- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline

  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 Table 6 : CPU Power impact \\
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 Table 6 : CPU Power impact \\


@@ -614,12 +622,11 @@ Table 6 : CPU Power impact \\
 %\label{overflow}}
 \end{figure}
 
 %\label{overflow}}
 \end{figure}
 

-Using the SIMGRID simulator flexibility, we have tried to determine the
+Using the Simgrid simulator flexibility, we have tried to determine the

 impact on the algorithms performance in varying the CPU power of the
 clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6
 confirm the performance gain, around 95\% for both of the two methods,
 impact on the algorithms performance in varying the CPU power of the
 clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6
 confirm the performance gain, around 95\% for both of the two methods,

-after adding more powerful CPU. Note that the execution time axis in the
-figure is in logarithmic scale.
+after adding more powerful CPU. 

 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}
 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}


@@ -627,12 +634,12 @@ Multisplitting algorithms in asynchronous mode}
 The previous paragraphs put in evidence the interests to simulate the
 behavior of the application before any deployment in a real environment.
 We have focused the study on analyzing the performance in varying the
 The previous paragraphs put in evidence the interests to simulate the
 behavior of the application before any deployment in a real environment.
 We have focused the study on analyzing the performance in varying the

-key factors impacting the results. In the same line, the study compares
-the performance of the two proposed methods in \textbf{synchronous mode
-}. In this section, with the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textbf{
-asynchronous mode} compare with the classical GMRES staying in the
-synchronous mode.
+key factors impacting the results. The study compares
+the performance of the two proposed algorithms both in \textit{synchronous mode
+}. In this section, following the same previous methodology, the goal is to
+demonstrate the efficiency of the multisplitting method in \textit{
+asynchronous mode} compared with the classical GMRES staying in
+\textit{synchronous mode}.

 
 Note that the interest of using the asynchronous mode for data exchange
 is mainly, in opposite of the synchronous mode, the non-wait aspects of
 
 Note that the interest of using the asynchronous mode for data exchange
 is mainly, in opposite of the synchronous mode, the non-wait aspects of


@@ -642,11 +649,10 @@ calculation without waiting for the end of the communication. Thus, the
 asynchronous may theoretically reduce the overall execution time and can
 improve the algorithm performance.
 
 asynchronous may theoretically reduce the overall execution time and can
 improve the algorithm performance.
 

-As stated supra, SIMGRID simulator tool has been used to prove the
+As stated supra, Simgrid simulator tool has been used to prove the

 efficiency of the multisplitting in asynchronous mode and to find the
 best combination of the grid resources (CPU, Network, input matrix size,
 efficiency of the multisplitting in asynchronous mode and to find the
 best combination of the grid resources (CPU, Network, input matrix size,

-\ldots ) to get the highest "\,relative gain" in comparison with the
-classical GMRES time.
+\ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.

 
 
 The test conditions are summarized in the table below : \\
 
 
 The test conditions are summarized in the table below : \\


@@ -656,11 +662,11 @@ The test conditions are summarized in the table below : \\
 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline

- Processors & 1 GFlops to 1.5 GFlops\\
-   Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
-   Inter-Network & bw=5 Mbits - lat=2E-02\\
+ Processors Power & 1 GFlops to 1.5 GFlops\\
+   Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
+   Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\

  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline

- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+ Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\

  \end{tabular}
 \end{footnotesize}
 
  \end{tabular}
 \end{footnotesize}
 


@@ -668,12 +674,9 @@ Again, comprehensive and extensive tests have been conducted varying the
 CPU power and the network parameters (bandwidth and latency) in the
 simulator tool with different problem size. The relative gains greater
 than 1 between the two algorithms have been captured after each step of
 CPU power and the network parameters (bandwidth and latency) in the
 simulator tool with different problem size. The relative gains greater
 than 1 between the two algorithms have been captured after each step of

-the test. Table I below has recorded the best grid configurations
-allowing a multiplitting method time more than 2.5 times lower than
-classical GMRES execution and convergence time. The finding thru this
-experimentation is the tolerance of the multisplitting method under a
-low speed network that we encounter usually with distant clusters thru the
-internet.
+the test. Table 7 below has recorded the best grid configurations
+allowing the multisplitting method execution time more performant 2.5 times than
+the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.

 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}


@@ -683,55 +686,33 @@ internet.
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 

+

 \begin{table}[!t]
   \centering
 \begin{table}[!t]
   \centering

-  \caption{Relative gain of the multisplitting algorithm compared with
-the classical GMRES}
-  \label{"Table 7"}
-
-  \begin{mytable}{6}
-    \hline
-    bandwidth (Mbit/s)
-    & 5         & 5         & 5         & 5         & 5 \\
-    \hline
-    latency (ms)
-    & 20      & 20      & 20      & 20      & 20 \\
-    \hline
-    power (GFlops)
-    & 1         & 1         & 1         & 1.5       & 1.5 \\
-    \hline
-    size (N)
-    & 62        & 62        & 62        & 100       & 100 \\
-    \hline
-    Precision
-    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\
-    \hline
-    Relative gain
-    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 \\
-    \hline
-  \end{mytable}
-
-  \smallskip
+%  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+%  \label{"Table 7"}
+Table 7. Relative gain of the multisplitting algorithm compared with
+the classical GMRES \\

 
 

-  \begin{mytable}{6}
+  \begin{mytable}{11}

     \hline
     bandwidth (Mbit/s)
     \hline
     bandwidth (Mbit/s)

-    & 50        & 50        & 50        & 50        & 50 \\
+    & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\

     \hline
     latency (ms)
     \hline
     latency (ms)

-    & 20      & 20      & 20      & 20      & 20 \\
+    & 20      & 20      & 20      & 20      & 20 & 20      & 20      & 20      & 20      & 20 \\

     \hline
     power (GFlops)
     \hline
     power (GFlops)

-    & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
+    & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\

     \hline
     size (N)
     \hline
     size (N)

-    & 110       & 120       & 130       & 140       & 150 \\
+    & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\

     \hline
     Precision
     \hline
     Precision

-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
+    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\

     \hline
     Relative gain
     \hline
     Relative gain

-    & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
+    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\

     \hline
   \end{mytable}
 \end{table}
     \hline
   \end{mytable}
 \end{table}